Multiplicity results for nonlocal elliptic transmission problem with variable exponent
DOI:
https://doi.org/10.5269/bspm.v33i2.23875Keywords:
Variational method, multiple solutions, nonlocal elliptic transmission problem, Mountain Pass Theorem, Ekeland's principleAbstract
In this paper, a transmission problem given by a system of two nonlinear equations of p(x)-Kirchho type with nonstandard growth conditions are studied. Using the mountain pass theorem combined with the Ekeland's variational principle, we obtain at least two distinct, non-trivial weak solutions.References
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2. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 04(1973) 349-381.
3. S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 3(2005) 515-545.
4. S.N. Antontsev and J.F. Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52(7)(2006) 19-36. http://dx.doi.org/10.1007/s11565-006-0002-9.
5. B. Cekic and R. A. Mashiyev, Nontrivial Solution for a nonlocal elliptic transmission problem in variable exponent Sobolev Spaces, Abstract and Applied Analysis Volume 2010, Article ID 385048, 12 pages, doi:10.1155/2010/385048.
6. F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods, Bulletin of the Australian Mathematical Society, vol. 74(2)(2006) 263-277.
7. G. Dai and R. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, Journal of Mathematical Analysis and Applications, vol. 359(1)(2009) 275-284.
8. G. Dai and D. Liu, Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, Journal of Mathematical Analysis and Applications, vol. 359(2)(2009.) 704-710
9. L. Diening, Theorical and numerical results for electrorheological fluids, Ph. D. Thesis, University of Freiburg, Germany (2002).
10. X. L.Fan & D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263(2001) 424-446.
11. X. L.Fan, J. S. Shen & D. Zhao, Sobolev embedding theorems for spaces W m,p(x) (Ω), J. Math. Anal. Appl. 262(2001) 749-760.
12. X. L. Fan and Q. H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Analysis: Theory, Methods & Applications, vol. 52(8)(2003) 1843-1852.
13. X. L. Fan, On nonlocal p(x)-Laplacian Dirichlet problems, Nonlinear Anal., 72(2010) 3314-3323.
14. T. C. Halsey, Electrorheological fluids, Science, 258(1992 761-766. http://dx.doi.org/10.1126/science.258.5083.761.
15. O. KováÄik & J. Rakosník, On spaces Lp(x) and Wk,p(x) , Czechoslovak Math. J. 41(1991) 592-618.
16. T. F. Ma and J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Applied Mathematics Letters, vol. 16 (2)(2003) 243-248.
17. A. Marzocchi, J. E. Muñoz Rivera, and M. G. Naso, Transmission problem in thermoelasticity with symmetry, IMA Journal of Applied Mathematics, vol. 68 (1)(2003) 23-46.
18. R.A. Mashiyev, S. Ogras, Z. Yucedag, and M. Avci, The Nehari manifold approach for Dirichlet problem involving the p(x)-Laplacian equation, J. Korean Math. Soc., 47(4)(2010) 845-860.
19. J. E. Munoz Rivera and H. P. Oquendo, The transmission problem of viscoelastic waves, Acta Applicandae Mathematicae, vol. 62(1)(2000) 1-21.
20. J. Y. Park, J. J. Bae, and I. H. Jung, Uniform decay of solution forwave equation of Kirchhoff type with nonlinear boundary damping and memory term, Nonlinear Analysis: Theory, Methods & Applications, vol. 50 (7)(2002) 871-884.
21. K. Pflüger, Nonlinear transmission problems in bounded domains of R n, Applicable Analysis, vol.62 (3-4)(1996) 391-403.
22. M. RÃ¥uzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000.
23. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Mathematics of the USSR-Izvestiya, vol. 9(4)(1987) 33-66.
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2014-08-19
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